General conditions for global stability in a single species population-toxicant model
Introduction
Let T(t), U(t) and N(t) represent at time t the concentration of toxicant in the environment, the concentration of the toxicant in the total population and the population biomass, respectively. Mass balance arguments drive to the following model:where is a sufficiently smooth functional such that the initial value problem given by (1) and T(0)⩾0, U(0)=kN(0), N(0)⩾0 admits a unique and continuable solution for all positive time.
This model, when the functional is given byis due to Freedman and Shukla [8].
The roots of model (1) in the biomathematics literature are quite deep. In the eighties a deterministic approach to the problem of developing mathematical modelling to assess the effects of pollutants on ecological system was proposed by Hallam and his coworkers [11], [12]. Their model is nowadays classic. It takes into account of three state variables (expressed as biomass or concentrations): a population, a toxicant in the environment (external toxicant) and the toxicant in the total population (internal toxicant). The toxicants dynamics are founded on mass balance equation. The population dynamics is assumed to be logistic whereas the coupling between the population and the toxicant is accomplished by a linear dose–response function, that is the population growth rate is assumed to be a linear function of the internal toxicant.
The key modelling feature of this model stands in the equations ruling the toxicant dynamics. In fact, the same toxicant dynamics has been proposed with different population equations (see e.g. [11], [12], [23]) and has been coupled to single and multi species problems in several contexts as Lotka Volterra and chemostat-like environment [5], [18].
Ordinary, delay-differential and stochastic models based on the Hallam's modelling idea can be found in the literature (e.g. [4], [6], [9]) and the ordinary basic model itself, or its slightly modified versions, is still subject of investigations (see e.g. [23]).
In the Hallam's model, the internal toxicant is defined with respect to the biomass of the total population, so that the internal toxicant dynamics does not explicitly depend on the density of the population.
In 1991 Freedman and Shukla [8] proposed a different version, model , , which is based on the consideration that the internal toxicant should be defined with respect to the total mass or volume of the environment where the population lives. This consideration, which makes the model of the ecotoxicological problem more visible, has been taken into account also in [1], where an extention to the spatial structured case is presented, resulting in a diffusive–convective model. A diffusion model is also considered in [2].
The papers dealing with these models usually provide results on the qualitative properties of solutions. In this direction, the stability analysis plays an important role. The nature of equilibria may be interpreted in terms of population survival or extinction, though different approaches based on the concepts of persistence and permanence may also give an account on the population fate (see e.g. [19]).
In this paper, we deal with the global stability for the population-toxicant model proposed by Freedman and Shukla. We allow the population dynamics to be ruled by a generic functional and find conditions that such functional have to satisfy in order to guarantee the global stability of the unique nontrivial equilibrium. As a particular case, we consider a global stability problem solved in [8] by means of the Lyapunov direct method and improve the conditions there obtained.
We make use of a geometric approach to the global stability analysis for ordinary differential equations emerged from a series of papers in the middle of 1990s (see [17] and the references contained therein) and recently applied to several epidemic models [3], [14], [16], [24].
Such method is based on the use of a higher-order generalization of the Bendixson's criterion. The existence of a robust Bendixson criterion for the system at hand, associated to the existence and unicity of an equilibrium, determines the global dynamics of the system. The main theorem of the method requires the use of a Lozinskiı̆ logarithmic norm.
The paper is organized as follows: in Section 2 the mathematical framework for proving the global stability is outlined. In Section 3, the population-toxicant model is introduced and sufficient conditions for the global stability of the unique nontrivial equilibrium are obtained in terms of the functional describing the population dynamics. In Section 4, the result is specialized for the logistic-like population dynamics proposed by Freedman and Shukla. Numerical verifications, useful to better understand the relationships between the different parameter restrictions are presented in Section 5.
Section snippets
General framework
In the next sections, we will introduce the population-toxicant model and will investigate its global dynamics. We will apply the general method developed in [17]: we think useful to briefly outline it here though short descriptions may be found also elsewhere (e.g [3], [24]).
Consider the autonomous dynamical systemwhere , open set and f∈C1(D). Let be an equilibrium of (3), i.e. . We recall that is said to be globally stable in D if it is locally stable and all
The population-toxicant model
Let us now consider system (1), say:where T(t), U(t) and N(t) represent, respectively, the concentration of toxicant in the environment, the concentration of the toxicant in the total population and the population biomass at time t. The meaning of the positive constant parameters is the following: Q is the toxicant input rate; δ0 and δ1 are the toxicant depletion rates in the environment and population, respectively; α is the depletion rate due to
A logistic-like model
Freedman and Shukla [8] consider the functional (2), say:where r(U) represents the growth rate coefficient and K(T) is the population carrying capacity. Both of them are not constant, being affected the former by the concentration of the toxicant in the population and the latter by the concentation of the toxicant in the environment. It is also requested that:andThese assumptions mean
Numerical investigations
In order to better clarify the relationship between conditions , , , , , , , , , , , , , , we present several examples of solutions of the logistic-like model discussed above, i.e. the model (1) where and r(U), K(T) verify , .
Assume the growth rate r(U) and the carrying capacity K(T) to be represented by the simplest functions satisfying , :In this specific case, the threshold toxicant values for which the population cannot reproduce or grow are
Conclusion
We have dealt with the global stability problem for the nontrivial equilibrium of the population-toxicant interaction model (1), where the population dynamics is ruled by a generic functional . The main result is Theorem 2 which establishes conditions on to get global stability for the unique nontrivial equilibrium of (1). These conditions do not preclude, in principle, strong nonlinearity of the functional (and hence of the equation governing the population dynamics).
Theorem 2 is proved
Acknowledgements
The present work has been performed under the auspices of the Italian National Group for the Mathematical Physics (GNFM-Indam). The authors are thankful to Prof. S. Rionero and Prof. C. Tebaldi for helpful hints and discussions. Thanks are also given to the anonimous referee for his/her useful comments.
The authors like also to pin that this work started while the authors were attending the Summer School on Mathematical Physics. Ravello, Italy, September 2002.
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